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Examine the consistency of the system of equation \(3x - y - 2z = 2;\,\,2y - z = - 1;3x - 5y = 3\)
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Examine the consistency of the system of equation \(3x - y - 2z = 2;\,\,2y - z = - 1;3x - 5y = 3\)

Answer

Matrix form of given equations is AX = B
\(\Rightarrow \left[ {\begin{array}{*{20}{c}} 3&{ - 1}&{ - 2} \\ 0&2&{ - 1} \\ 3&5&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 3 \end{array}} \right]\)
Here \(A = \left[ {\begin{array}{*{20}{c}} 3&{ - 1}&{ - 2} \\ 0&2&{ - 1} \\ 3&5&0 \end{array}} \right]\)
\(\therefore \left| A \right| = \left| {\begin{array}{*{20}{c}} 3&{ - 1}&{ - 2} \\ 0&2&{ - 1} \\ 3&{ - 5}&0 \end{array}} \right|\)
\(\Rightarrow \left| A \right| = 3\left( {0 - 5} \right) - \left( { - 1} \right)\left( {0 + 3} \right) + \left( { - 2} \right)\left( {0 - 6} \right)\)\(= 3\left( { - 5} \right) + 3 + 12 = - 15 + 15 = 0\)
Now \(\left( {adj.A} \right) = \left[ {\begin{array}{*{20}{c}} { - 5}&{10}&5 \\ { - 3}&6&3 \\ { - 6}&{12}&6 \end{array}} \right]\) 
And \(\left( {adj.A} \right)B = \left[ {\begin{array}{*{20}{c}} { - 5}&{10}&5 \\ { - 3}&6&3 \\ { - 6}&{12}&6 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 3 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 10 - 10 + 15} \\ { - 6 - 6 + 9} \\ { - 12 - 12 + 18} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 5} \\ { - 3} \\ { - 6} \end{array}} \right] \ne 0\)
Therefore, given equations are inconsistent.
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